Answer :
To determine the population of a town after 5 years, given an initial population of 10,000 and an annual growth rate of 5%, we use the concept of exponential growth. Here's a step-by-step explanation of the solution:
1. Identify the initial population and growth rate:
- Initial population ([tex]\( P_0 \)[/tex]): 10,000
- Annual growth rate ([tex]\( r \)[/tex]): 5% or 0.05 (as a decimal)
2. Determine the number of years the population grows:
- Number of years ([tex]\( t \)[/tex]): 5
3. Formula for exponential growth:
The population after [tex]\( t \)[/tex] years can be calculated using the compound interest formula:
[tex]\[ P(t) = P_0 \times (1 + r)^t \][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population after [tex]\( t \)[/tex] years
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( r \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the number of years
4. Substitute the given values into the formula:
[tex]\[ P(5) = 10000 \times (1 + 0.05)^5 \][/tex]
[tex]\[ P(5) = 10000 \times (1.05)^5 \][/tex]
5. Calculate [tex]\( (1.05)^5 \)[/tex]:
[tex]\[ (1.05)^5 = 1.2762815625 \][/tex]
6. Multiply the initial population by the growth factor:
[tex]\[ P(5) = 10000 \times 1.2762815625 \][/tex]
[tex]\[ P(5) = 12762.815625 \][/tex]
7. Round the population to the nearest whole number:
[tex]\[ P(5) \approx 12763 \][/tex]
Thus, the population of the town after 5 years, rounded to the nearest whole number, will be 12,763.
1. Identify the initial population and growth rate:
- Initial population ([tex]\( P_0 \)[/tex]): 10,000
- Annual growth rate ([tex]\( r \)[/tex]): 5% or 0.05 (as a decimal)
2. Determine the number of years the population grows:
- Number of years ([tex]\( t \)[/tex]): 5
3. Formula for exponential growth:
The population after [tex]\( t \)[/tex] years can be calculated using the compound interest formula:
[tex]\[ P(t) = P_0 \times (1 + r)^t \][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population after [tex]\( t \)[/tex] years
- [tex]\( P_0 \)[/tex] is the initial population
- [tex]\( r \)[/tex] is the growth rate
- [tex]\( t \)[/tex] is the number of years
4. Substitute the given values into the formula:
[tex]\[ P(5) = 10000 \times (1 + 0.05)^5 \][/tex]
[tex]\[ P(5) = 10000 \times (1.05)^5 \][/tex]
5. Calculate [tex]\( (1.05)^5 \)[/tex]:
[tex]\[ (1.05)^5 = 1.2762815625 \][/tex]
6. Multiply the initial population by the growth factor:
[tex]\[ P(5) = 10000 \times 1.2762815625 \][/tex]
[tex]\[ P(5) = 12762.815625 \][/tex]
7. Round the population to the nearest whole number:
[tex]\[ P(5) \approx 12763 \][/tex]
Thus, the population of the town after 5 years, rounded to the nearest whole number, will be 12,763.
